Chicago's politically embattled Mayor Harold Washington tonight told the city that he is going to have to lay off 569 employes, eliminate a pending $22 million tax cut and take other cost-cutting steps to balance a $59 million budget deficit.

In a speech carried live by several local television and radio stations, Washington said that if the City Council, which he does not control, resists his plan he will dismiss 2,000 city workers beginning next month to achieve the savings he says are necessary.

"I am cognizant of the council's need for thoughtful deliberations before taking action, but the clock is steadily ticking," Washington said. "The longer we delay, the more severe the measures that will be required. If the ordinance reinstating the $22 million tax is not passed by Aug. 17, I will be compelled to lay off an additional 2,000 people from the city work force, effective Sept. 1. So I urge the council to act with dispatch."

Washington, Chicago's first black mayor, has been on the defensive since 29 white aldermen took control of the 50-member City Council his first day in office in May. They ousted his supporters from key committee chairmanships and opposed many of his proposals.

Tonight's speech was seen as a major attempt for Washington to break out of the impasse, go over the heads of his opponents and carry his message to the citizens.

Layoffs of the scale Washington proposed tonight are nearly unheard of in Chicago, whose 40,000-strong municipal labor force is the key to the longtime Democratic machine, which controls the city jobs through ward committeemen.

As one of his close aides said, the mayor's strategy in dealing with his budget problems "is a carrot-and-stick approach.

Led by Cook County Democratic Chairman Edward R. (Fast Eddie) Vrdolyak, who also is a city alderman, Washington's opponents are expected to recommend furloughs instead of layoffs.

Vrdolyak's chief lieutenant, Alderman Edward Burke, a former policeman and now a lawyer, opposed Washington's tax increase. "I do not believe a tax increase is justified, warranted, or tolerable," Burke said in a televised rebuttal speech. He questioned the mayor's budget figures, saying his own estimates showed a deficit of not more than $20 million.

The $22 million in taxes in question was first lifted by Wahington's predecessor, Jane M. Byrne, to bolster her chances for renomination before the Democratic primary in February. She lost, but the tax relief stayed in place, and is scheduled to show up in tax bills in August, 1984.

"Our city stands in the midst of a financial problem that can only be solved by immediate and drastic action," Washington said in a 15-minute speech. "If we do not act, the city could face a severe fiscal crisis."

He said that if Chicago is to be solvent "We will not be able to accomplish these goals without calling for sacrifices by thousands of city employes, and by you, the citizens."

He blamed the deficit on "national economic policies which have taken a terrible toll of big cities like Chicago. And some of it is plainly the result of lax management at City Hall."

The recession, a steady flight of better-off white residents to the suburbs and reduced federal aid to the city have combined to take their toll of its finances, he said.

But the mayor's budget chief, David Schulz, also provided figures that showed that while the city's population had declined to 3 million from 3.4 million between 1970 and today, the municipal work force had grown in the same period from 40,702 to 41,480.

"The tax base of the city simply cannot support the current level of expenditures and employes, either of the remainder of 1983 or in 1984 and beyond."

Washington also wants to sell three city-owned-and-run parking garages in the downtown, for a net of $11 million.

The politicking around the city's financial condition is being watched closely by bankers and municipal financial analysts. Chicago must borrow up to $100 million at the end of the year to pay bills and carry payroll and other obligations through early 1984 until property tax receipts start rolling in.

At present, the city has an A-minus rating, down from the halcyon years of high employment, a booming economy, and balanced budgets of the late Richard J. Daley in the 1960s and 1970s.

Washington already has banned new city hiring, canceled back-pay increases, restricted overtime and reduced out-of-state travel by city officials. "We simply must discipline ourselves to live within our means," he said.

Among those to be laid off will be clerical workers in the police and fire departments, and health, streets and public works employes. Hundreds of sewer workers will be put on four-day weeks through 1983. The sewer department is one of the most politicized in the city; Byrne padded its payroll with hundreds of workers before she left office. Washington said a new management team will professionalize the department.

In his rebuttal, Burke made clear that there is hard slogging ahead for the mayor in gaining agreement from the council majority. While noting that "about 60 percent" of the mayor's plan is in agreement with the majority's views, he bridled at Washington's threat of firing 2,000 if the council withholds approval.

"It's a sword of Damocles," Burke said. CAPTION: Picture, Mayor Washington measures aimed at balancing projected $59 million budget deficit. By Harry Naltchayan--The Washington Post